How to Calculate the Confidence Interval of the Mean

By Mark Kennan; Updated April 24, 2017

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The confidence interval of the mean is a statistical term used to describe the range of values in which the true mean is expected to fall, based on your data and confidence level. The most commonly used confidence level is 95 percent, meaning that there is a 95 percent probability that the true mean lies within the confidence interval you've calculated. To calculate the confidence interval, you need to know the mean of your data set, the standard deviation, the sample size and your chosen confidence level.

Calculate the mean, if you haven't done so already, by adding all of the values in your data set and dividing by the number of values. For example, if your data set were 86, 88, 89, 91, 91, 93, 95 and 99, you would get 91.5 for the mean.

Calculate the standard deviation for the data set, if you haven't done so already. In our example, the standard deviation of the data set is 4.14.

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Determine the standard error of the mean by dividing the standard deviation by the square root of the sample size. In this example, you would divide 4.14, the standard deviation, by the square root of 8,the sample size, to get about 1.414 for the standard error.

Determine the critical value for t by using a t-table. You can find one in your statistics textbook or via an online search. The number of degrees of freedom is equal to one less than the number of data points in your set -- in our case, 7 -- and the p-value is the confidence level. In this example, if you wanted a 95 percent confidence interval and you had seven degrees of freedom, your critical value for t would be 2.365.

Multiply the critical value by the standard error. Continuing the example, you would multiply 2.365 by 1.414 and get 3.344.

Subtract this figure from the mean of your data set, and then add this figure to the mean, to find the lower and upper limit of the confidence interval. For example, you would subtract 3.344 from the mean of 91.5 to find the lower limit to be 88.2, and add it to find the upper limit to be 94.8. This range, 88.2 to 94.8, is your confidence interval for the mean.

Tip

If you need to brush up on how to calculate the standard deviation of your data set, the information is easily found online or in your statistics textbook.

References

About the Author

Mark Kennan is a writer based in the Kansas City area, specializing in personal finance and business topics. He has been writing since 2009 and has been published by "Quicken," "TurboTax," and "The Motley Fool."